### 数学代写|计量经济学原理代写Principles of Econometrics代考|Probability Primer

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Random Variables

Benjamin Franklin is credited with the saying “The only things certain in life are death and taxes.” While not the original intent, this bit of wisdom points out that almost everything we encounter in life is uncertain. We do not know how many games our football team will win next season. You do not know what score you will make on the next exam. We don’t know what the stock market index will be tomorrow. These events, or outcomes, are uncertain, or random. Probability gives us a way to talk about possible outcomes.

A random variable is a variable whose value is unknown until it is observed; in other words, it is a variable that is not perfectly predictable. Each random variable has a set of possible values it can take. If $W$ is the number of games our football team wins next year, then $W$ can take the values $0,1,2, \ldots, 13$, if there are a maximum of 13 games. This is a discrete random variable since it can take only a limited, or countable, number of values. Other examples of discrete random variables are the number of computers owned by a randomly selected household, and the number of times you will visit your physician next year. A special case occurs when a random variable can only be one of two possible values-for example, in a phone survey, if you are asked if you are a college graduate or not, your answer can only be “yes” or “no.” Outcomes like this can be characterized by an indicator variable taking the values one if yes or zero if no. Indicator variables are discrete and are used to represent qualitative characteristics such as sex (male or female) or race (white or nonwhite).

The U.S. GDP is yet another example of a random variable, because its value is unknown until it is observed. In the third quarter of 2014 it was calculated to be $16,164.1$ billion dollars. What the value will be in the second quarter of 2025 is unknown, and it cannot be predicted perfectly. GDP is measured in dollars and it can be counted in whole dollars, but the value is so large that counting individual dollars serves no purpose. For practical purposes, GDP can take any value in the interval zero to infinity, and it is treated as a continuous random variable. Other common macroeconomic variables, such as interest rates, investment, and consumption, are also treated as continuous random variables. In finance, stock market indices, like the Dow Jones Industrial Index, are also treated as continuous. The key attribute of these variables that makes them continuous is that they can take any value in an interval.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Probability Distributions

Probability is usually defined in terms of experiments. Let us illustrate this in the context of a simple experiment. Consider the objects in Table P.1 to be a population of interest. In statistics and econometrics, the term population is an important one. A population is a group of objects, such as people, farms, or business firms, having something in common. The population is a complete set and is the focus of an analysis. In this case the population is the set of ten objects shown in Table P.1. Using this population, we will discuss some probability concepts. In an empirical analysis, a sample of observations is collected from the population of interest, and using the sample observations we make inferences about the population.

If we were to select one cell from the table at random (imagine cutting the table into 10 equally sized pieces of paper, stirring them up, and drawing one of the slips without looking), that would constitute a random experiment. Based on this random experiment, we can define several random variables. For example, let the random variable $X$ be the numerical value showing on a slip that we draw. (We use uppercase letters like $X$ to represent random variables in this primer). The term random variable is a bit odd, as it is actually a rule for assigning numerical values to experimental outcomes. In the context of Table P.1, the rule says, “Perform the experiment (stir the slips, and draw one) and for the slip that you obtain assign $X$ to be the number showing.” The values that $X$ can take are denoted by corresponding lowercase letters, $x$, and in this case the values of $X$ are $x=1,2,3$, or 4 .

For the experiment using the population in Table P.1, ${ }^{1}$ we can create a number of random variables. Let $Y$ be a discrete random variable designating the color of the slip, with $Y=1$ denoting

a shaded slip and $Y=0$ denoting a slip with no shading. The numerical values that $Y$ can take are $y=0,1$.

Consider $X$, the numerical value on the slip. If the slips are equally likely to be chosen after shuffling, then in a large number of experiments (i.e., shuffling and drawing one of the slips), $10 \%$ of the time we would observe $X=1,20 \%$ of the time $X=2,30 \%$ of the time $X=3$, and $40 \%$ of the time $X=4$. These are probabilities that the specific values will occur. We would say, for example, $P(X=3)=0.3$. This interpretation is tied to the relative frequency of a particular outcome’s occurring in a large number of experiment replications.

We summarize the probabilities of possible outcomes using a probability density function (pdf). The $p d f$ for a discrete random variable indicates the probability of each possible value occurring. For a discrete random variable $X$ the value of the $p d f f(x)$ is the probability that the random variable $X$ takes the value $x, f(x)=P(X=x)$. Because $f(x)$ is a probability, it must be true that $0 \leq f(x) \leq 1$ and, if $X$ takes $n$ possible values $x_{1}, \ldots, x_{n}$, then the sum of their probabilities must be one
$$f\left(x_{1}\right)+f\left(x_{2}\right)+\cdots+f\left(x_{n}\right)=1$$

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Marginal Distributions

Working with more than one random variable requires a joint probability density function. For the population in Table P.1 we defined two random variables, $X$ the numeric value of a randomly drawn slip and the indicator variable $Y$ that equals 1 if the selected slip is shaded, and 0 if it is not shaded.

Using the joint $p d f$ for $X$ and $Y$ we can say “The probability of selecting a shaded 2 is $0.10 . “$ This is a joint probability because we are talking about the probability of two events occurring simultaneously; the selection takes the value $X=2$ and the slip is shaded so that $Y=1$. We can write this as
$$P(X=2 \text { and } Y=1)=P(X=2, Y=1)=f(x=2, y=1)=0.1$$
The entries in Table P.3 are probabilities $f(x, y)=P(X=x, Y=y)$ of joint outcomes. Like the $p d f$ of a single random variable, the sum of the joint probabilities is 1 .

Given a joint $p d f$, we can obtain the probability distributions of individual random variables, which are also known as marginal distributions. In Table P.3, we see that a shaded slip, $Y=1$, can be obtained with the values $x=1,2,3$, and 4 . The probability that we select a shaded slip is the sum of the probabilities that we obtain a shaded 1, a shaded 2, a shaded 3 , and a shaded 4 . The probability that $Y=1$ is
$$P(Y=1)=f_{Y}(1)=0.1+0.1+0.1+0.1=0.4$$
This is the sum of the probabilities across the second row of the table. Similarly the probability of drawing a white slip is the sum of the probabilities across the first row of the table, and $P(Y=0)=f_{Y}(0)=0+0.1+0.2+0.3=0.6$, where $f_{Y}(y)$ denotes the $p d f$ of the random variable $Y$. The probabilities $P(X=x)$ are computed similarly by summing down across the values of $Y$. The joint and marginal distributions are often reported as in Table P.4. ${ }^{3}$

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Probability Distributions

F(X1)+F(X2)+⋯+F(Xn)=1

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## MATLAB代写

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